MITC shell benchmarks

Verification methodBENCHMARK-SPECIFIC · no transfer to every shell route

Cook, Scordelis–Lo, pinched cylinder, modal plate, transient, harmonic, and laminate cases probe different parts of the shell formulation.

01 Model and scope

The benchmark set probes different failure modes: bending-dominated response, transverse shear, curved geometry, modal mass/stiffness, time integration and harmonic response. No one benchmark validates all shell operators or every laminate and damping combination.

Route assumptions and calculation checks

Record element count, thickness, tying points, boundary conditions, integration, stress recovery, and mesh distortion.
Modal comparison uses subspace MAC for repeated square-plate frequencies.
External correlation is named by comparator and observable; it is not physical validation.

Useful foundations and prerequisites

Units, signs and error measures Interpolation, mapping and assembly

02 Essential equations and symbols

This sheet records a scope or execution boundary. Its detailed steps explain the relevant conditions.

Definitions and conventions

$w$: Transverse displacement observable where applicable.
$\omega{}_{i}$: Natural frequencies for modal cases.
u(t): Transient response history.
$\hat{u}(\omega{})$: Complex harmonic response.

01Named shell fixture

02Static / modal / dynamic

03Matched observables

04Route-specific evidence

MITC shell benchmarks · method mapEach named benchmark probes a different shell operator; comparisons stay tied to its geometry, output, and analysis route.

4 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Assign each shell benchmark a distinct question

Cook-type bending probes in-plane/bending response; Scordelis–Lo and the pinched cylinder probe curved-shell and shear behaviour; the plate case probes modal mass and stiffness; transient and harmonic cases probe their own time/frequency operators; laminate cases probe section coupling and ply recovery.

\mathbf K_e=\mathbf K_m+\mathbf K_b+\mathbf K_s+\mathbf K_d+\mathbf K_{mb}

2 · Freeze geometry, interpolation, and output definition

For each case, record element count, thickness, mesh distortion, tying locations, boundary conditions, integration, load, stress-recovery points, and the exact displacement or frequency observable. A changed choice can alter the reference problem, so benchmark labels alone are not enough.

e_{rel}=\frac{|y_{QF}-y_{ref}|}{|y_{ref}|},\qquad\Delta_{mesh}=\frac{|y_h-y_{h/2}|}{\max(|y_{h/2}|,s_0)}

3 · Compare subspaces for repeated modal roots

Symmetric square plates can have repeated or near-repeated modes. Compare the mass-weighted modal subspace instead of matching arbitrary eigenvector signs and order. The transient and harmonic results remain separate from this eigenvalue comparison.

\operatorname{MAC}_M(\phi_i,\psi_j)=\frac{|\phi_i^T\mathbf M\psi_j|^2}{(\phi_i^T\mathbf M\phi_i)(\psi_j^T\mathbf M\psi_j)}

4 · State what external correlation does not prove

Correlation with a published benchmark or another solver is numerical evidence for the named observable and setup. It is not a physical experiment, and no single shell benchmark validates every laminate, damping, material, dynamic, or mesh route.

04 Algorithm and implementation

This sheet provides explanatory references rather than an inline source excerpt. Use the labeled sources to inspect the precise boundary.

05 Verification and limitations

BENCHMARK-SPECIFIC · no transfer to every shell route

The formulation and its explanatory checks do not by themselves establish a qualified application domain. Consult the capability matrix and the applicable recorded case.

06 Sources and related sheets

Shell demonstrations MITC4 Newmark input MITC4 harmonic input

Continue the reasoning

TET4 force, pressure, and affine patch Beam and discrete oscillator J2 bar External solver comparison Recorded structured-TET4 workloads

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